How to find a point on a line within a fixed distance

It is well known that on a line, a target point in unknown position can be found by walking a path at most 9 times as long as the distance from the start to the target point, in the worst case. This competitive factor of 9 is optimal. We investigate the case where the target is known to be within a fixed distance, r, of the start point, and determine the optimum competitive factor, C(r) < 9, that can be achieved by a competitive strategy S(r), under this additional assumption.